The proposer intends to continue his recent research on the mathematical analysis of several models of chemical and biochemical systems. These models include the Fitzhugh's equations of nerve conduction, the Glass-Kauffman model of cellular dynamics and Bell's model of Immune response in animals. The proposer has recently derived a criterion for the stability of bifurcating periodic solutions of autonomous, nonlinear systems. However, it only applies to the case that the criterion is not equal to zero. Thus, the proposer intends to derive a new criterion under this case and apply it to the Fitzhugh equations. The proposer has successfully applied his criterion to determine the stability of bifurcating periodic solutions of the Glass-Kauffman system of 2-cells model. The proposer has also proven that this 2-cells model has nonlocal periodic solution. It is proposed to extend these results to the N-cells model for N greater than 2, also to prove that the family of bifurcating solutions grows to become a large periodic solution and connects with the nonlocal periodic solution. The proposer intends to continue to investigate Bell's model by using both the bifurcation theorem and global analysis.